New York University, USA, adam.brandenburger@stern.nyu.edu
(joint work with Patricia Contreras-Tejada, Pierfrancesco La Mura, Giannicola Scarpa, and Kai Steverson)
The Agreement Theorem (Aumann, 1976) states that if two Bayesian agents start with a common prior, then they cannot have common knowledge that they hold different posterior probabilities of some underlying event of interest. In short, the two agents cannot “agree to disagree.” This result applies in the classical domain where classical probability theory applies. But in non-classical domains (such as the quantum world), classical probability theory does not apply, and so we cannot assume that the same result holds when agents observe non-classical phenomena. Inspired by their use in quantum mechanics (Wigner, 1932; Dirac, 1942; Feynman, 1987; Wooters, 1987), we employ signed probability measures (“quasi-probabilities”) to investigate the epistemics of the non-classical world and ask, in particular: What conditions lead to agreement or allow for disagreement when agents may use signed probabilities?
[1] Aumann, R., “Agreeing to Disagree,” Annals of Statistics, 4, 1976, 1236-1239.
[2] Dirac, P., “The Physical Interpretation of Quantum Mechanics,” Proceedings of the Royal Society of London (Series A: Mathematical and Physical Sciences), 180, 1942, 1-40.
[3] Feynman. R., “Negative Probability,” in B. Hiley and F. Peat (eds.), Quantum Implications: Essays in Honour of David Bohm, Routledge and Kegan Paul, 1987, 235-248.
[4] Wigner, E., “On the Quantum Correction for Thermodynamic Equilibrium,” Physical Review, 40, 1932, 749-759.
[5] Wooters, W., “A Wigner-Function Formulation of Finite-State Quantum Mechanics,” Annals of Physics, 176, 1987, 1-21.